Problem: A curve is defined by the parametric equations $x=2^t$ and $y=8^t$. What is $\dfrac{d^2y}{dx^2}$ in terms of $t$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $6\cdot2^t$ (Choice B) B $3\cdot4^t$ (Choice C) C $\dfrac{3\cdot2^t}{\ln(2)}$ (Choice D) D $4^t$
Solution: We are asked to find the second derivative of a parametric function. Recall that the first derivative of a function defined parametrically by the equations $x=u(t)$ and $y=v(t)$ is found with the following rule: $\dfrac{dy}{dx}=\dfrac{\left(\dfrac{dy}{dt}\right)}{\left(\dfrac{dx}{dt}\right)}=\dfrac{v'(t)}{u'(t)}$ Then, the second derivative is found with this following rule: $\dfrac{d^2y}{dx^2}=\dfrac{\dfrac{d}{dt}\left(\dfrac{dy}{dx}\right)}{\left(\dfrac{dx}{dt}\right)}=\dfrac{\dfrac{d}{dt}\left(\dfrac{v'(t)}{u'(t)}\right)}{u'(t)}$ Let's start by finding $\dfrac{dy}{dx}$. $\dfrac{dy}{dx}=3\cdot4^t$ Now we can find $\dfrac{d^2y}{dx^2}$. $\begin{aligned} \dfrac{d^2y}{dx^2}&=\dfrac{\dfrac{d}{dt}\left(\dfrac{dy}{dx}\right)}{\left(\dfrac{dx}{dt}\right)} \\\\ &=\dfrac{\dfrac{d}{dt}\left(3\cdot4^t\right)}{\dfrac{d}{dt}(2^t)} \\\\ &=\dfrac{3\cdot4^t\ln(4)}{2^t\ln(2)} \\\\ &=3\cdot2^t\cdot\dfrac{\ln(2^2)}{\ln(2)} \\\\ &=3\cdot2^t\cdot\dfrac{2\ln(2)}{\ln(2)} \\\\ &=6\cdot2^t \end{aligned}$ In conclusion, $\dfrac{d^2y}{dx^2}=6\cdot2^t$.